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Geostats Intro 1INTRODUCTION TO GEOSTATISTICSAmvrossios C. BagtzoglouDepartment of Civil and Environmental EngineeringUniversity of Connecticut1Geostats Intro 2Sources of Uncertainty•Measurements• Parameters• Models• Geometry & BCs or ICsGeostats Intro 3COURSE OUTLINE• PROBABILITY PRIMER– Stochastic Processes– Fourrier Transforms and Applications– Auto-correlation & Cross-correlation– Power Spectrum & Cross-Spectrum– Variogram1Geostats Intro 4HYDROLOGIC OBSERVATIONS AS STOCHASTIC PROCESSES1Geostats Intro 6( ) propability of( ) 1for a certain event( ) ( ) ( ) for mutually exclusive( ) lim (relative frequency approach)( ) (classical approach:possible outcomes)cAnAPA APSPA B PA PBnPAnNPAN→∞→/=∪= +==Probability PrimerGeostats Intro 7~where ( ) is assigned to everyis an outcomeRV xξξξ=The Cumulative Distribution Function (CDF)~() ( )xFx Px x x=≤ −∞≤≤∞Random Variables (RVs)Must satisfy:•Inverse mapping of the set of allxis an event•Probability of this event is zero, that is:~()0Px=±∞ =1Geostats Intro 8Properties of CDF[]12 1 2~1221~~1) ( ) 0()12) ( ) ( )3) ( ) 0 ( ) 04) ( ) 1 ( )5) ( ) ( ) ( )6) ( ) ( )is the PDF (probability density function))() 0)( ()7) () ()ooFFx x Fx FxFx Fx x xPx x FxPx x x Fx Fxdfx Fxdxafxbpx x x x fx xFx fxdx−∞ =∞=<⇒ ≤=⇒ =∀≤>=−≤≤ = −=≥≤≤+∆≈ ∆=∫Geostats Intro 9PDFsRV is Gaussian:22()21()2( , ); (0,1)is more commonxfx eRV N Nµσσπµσ−−==RV is uniform1221121()0&xxxxxfxxx xx∀≤≤−=∀< >Log-normal distribution(for non-negative parameters)ln( )gf=1Geostats Intro 10Moments of PDFs~~22 2 2~~~~~ ~~~~~~~~( ) ( ) (first moment)( ) ( ) (second central moment)Also known as ( )(, ) ( )( )(, )Correlation Coefficientif 0 thxyxyxEx xfxdxxx xfxdxVar xCov x y x y x y x yCov x yrCov rµσµµµσσ∞−∞∞−∞=< >= ==< > − < > = −=< − − >=< > − < >< >===→∫∫~~~~ere is no relationif 1 the samexy x yCov r→< >=< >< >==→Geostats Intro 11Stochastic (random) Processes~~~~~~~~~~~~()(,,) (,)ensemble/family of functions [ , , ]sample or unique realization[ , ]:fixRV is state variable :fix ,RV is scalar: fix , ,() timeprocess () () ()Auto-correlxRV ZRP Z x t Z x txtxtxtxtZt t Zt Zf tdZξξξξξξµ∞−∞→→→=< >=∫*12 1 2 12 12 1 212 1 2*ation (,) ()() (,) (, ;,)complex conjugateRt t Zt Zt Rt t ZZ f Z Z t t dZdZ=< >= =∫∫Autocovariance (One Process)*12 12 1 21212~(, ) (, ) () ()if(, ) [()]Ct t Rt t t tttCt t VarZtµµ=−==1Geostats Intro 12AUTO-CORRELATION AND CUTOFF1Geostats Intro 14Fourrier Transforms & Applicationsˆˆˆ2() (): angular velocity (rad/time)( ): auto-correlation functionInverse transform:1() ( )2or() ( ) ( 2 ): frequencyiiifSRedRtRSedR S f e df ffωτωτπτωττωτωωπτωπ∞−−∞∞−∞∞−−∞====∫∫∫Spectrum: (Power spectrum, spectral density) of a process Z(t)Geostats Intro 15Cross Spectrumˆˆ**ˆ**() ()1() ( )2() ()() ( )1() ( ) ( )2igh ghigh ghhgghigh ghgh hg ghSRedRSedRRRRRR R R S edωτωτωτωτττωωπττττττ ωωπ∞−−∞∞−∞∞−∞==−==−≠⇒ = −=∫∫∫whereas S(ω)is real, regardless of process Z, Sghis (in general) complex (even if g, hare real)Generalized Spectrumˆ() () ()spatial vectorwave # vectorikdSk Gk R e dkξξξξ−∞−−−−∞−====∫1Geostats Intro 16If a stochastic process is statistically homogeneous it will always have a spectrum.Can use representation theorem:ˆ3could be used interchangeably1() ()(2 )(), () , ()ikxyygx e dZ kdZ k G k d k S kπ−−−−−−−−=∫∫∫Geostats Intro 17Examples of Signal, Auto-correlation, and Spectrum1Geostats Intro 18COURSE OUTLINE• VARIOGRAPHY–Basics– Discussion of Common Variogram Models– Directional and Zonal Variography– Example• KRIGING– Methods of Spatial Interpolation– Simple Kriging– Ordinary Kriging– Universal Kriging (more details later)– Block KrigingGeostats Intro 19COURSE OUTLINE• TEST CASE– Application of Geostatistics for Enhancing Resistivity Data2Geostats Intro 20VARIOGRAPHYGeostats Intro 21VariographyIntroduction to Variography– Basic Introduction– Lab exercise– Distribution of Software3Geostats Intro 22Introduction to Variography• What is a variogram?• Why are variograms needed?• How are variograms made?• Variogram structure• Variogram models• Example and/or Lab exerciseGeostats Intro 23What is a Variogram?•A variogramis a spatial decomposition of a data set or a partition of sample variance•Summarizes the relationship between DIFFERENCESof pairs of measurements and the distance between them4Geostats Intro 24Example: Rainfall over a regionGeostats Intro 25Traditional Summary Statistics• Mean – arithmetic, geometric• Median – 0.50 quantile• Mode – Histogram peak• Variance (mean square difference from mean)– Standard deviation, s•Skewness ks– evaluates symmetryuses cubic difference from mean, weighted by s35Geostats Intro 26Why are Variograms needed?Traditional Statistical Methodssuppress spatial informationGeostats Intro 27Why are Variograms needed?•Sample Variance S = 1/(N-1) Σ(i=1 to N)(Zi-Zm)2Can also be written:S = 1/2N(N-1) Σ(i=1 to N)Σ(j=1 to N)(Zi-Zj)2Where square difference of all pairs areplaced in the samebin and averaged6Geostats Intro 28Why are Variograms needed?The variogram divides the squared differences in DISTINCT bins based on the distance between the corresponding points: γ* (h) = 1/(2Nh) Σ(I,j ∈Ph to Nh)(Zi-Zj)2Where Nh= Number of distinct data pairsPlaced in the set Ph(pairs displaced by h)Geostats Intro 29Variogram FunctionFor intrinsic random functions:γ(h) = 0.5 Var [Z(x+h)-Z(x)]For stationary and intrinsic variables:Mean of random functions: Z(x+h)-Z(x) = 0 ; thereforeγ(h) = 0.5 E [Z(x+h)-Z(x)]27Geostats Intro 30Variogram Details• Stationarity: variable distribution (and moments such as mean and variance) are invariant under translation• Intrinsic hypothesis: Assumes increments of a function (random field) are ‘weakly’ stationary:The mean and variance of the increments ofZ(x+h)-Z(x) exist and are independent of the point xWhat does this mean for trending data?Geostats Intro 31Spatial Covariance• C(0) = σ2• C(h) = C(-h)•|C(h)| <C(0)γ(h) = C(0) - C(h)8Geostats Intro 32Variogram AppearanceGeostats Intro 33Typical Variogram Structures9Geostats Intro 34Common Variogram Models•Linear–Power•Spherical• Exponential• Gaussian•


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